Questions and Answers
What is the point slope form?
What is the slope intercept form?
Define the difference of cubes.
x³ - y³ = (x - y)(x² + xy + y²)
Define the sum of cubes.
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What is the quadratic formula?
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What is the standard form of a quadratic equation?
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What is the standard form for the equation of a circle?
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Define the difference of squares.
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What is the representation of imaginary numbers?
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Define the distance formula.
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What is the midpoint formula?
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What condition must hold for a graph to be symmetric with respect to the x-axis?
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What condition must hold for a graph to be symmetric with respect to the y-axis?
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What condition must hold for a graph to be symmetric with respect to the origin?
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Define completing the square.
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What is the formula for the slope of a line passing through two points?
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What condition indicates that two lines are parallel?
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What condition indicates that two lines are perpendicular?
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Define an even function.
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Define an odd function.
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What is the greatest integer function?
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What is a vertical shift upward in terms of a function?
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What is a vertical shift downward in terms of a function?
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What is a horizontal shift to the right in terms of a function?
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What is a horizontal shift to the left in terms of a function?
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What represents a reflection in the x-axis?
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What represents a reflection in the y-axis?
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How is the sum of two functions represented?
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How is the difference of two functions represented?
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How is the product of two functions represented?
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How is the quotient of two functions represented?
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How is the composition of two functions represented?
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Study Notes
Algebra II Formulas
-
Point Slope Form:
Useful for writing the equation of a line given a point ((x_1, y_1)) and the slope (m). Formula: (y - y_1 = m(x - x_1)). -
Slope Intercept Form:
Represents linear equations, where (m) is the slope and (b) the y-intercept. Formula: (y = mx + b). -
Difference of Cubes:
A polynomial identity used to factor the difference of two cubes. Formula: (x^3 - y^3 = (x - y)(x^2 + xy + y^2)). -
Sum of Cubes:
A polynomial identity for factoring the sum of two cubes. Formula: (x^3 + y^3 = (x + y)(x^2 - xy + y^2)). -
Quadratic Formula:
Used to find the roots of a quadratic equation (ax^2 + bx + c = 0). Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). -
Quadratic Equation:
A second-degree polynomial equation expressed as (ax^2 + bx + c = 0). -
Standard Form for Equation of a Circle:
Represents a circle's equation, where ((m,n)) is the center and (r) is the radius. Formula: ((x - m)^2 + (y - n)^2 = r^2). -
Difference of Squares:
A factoring identity for the difference of two squares. Formula: (x^2 - y^2 = (x + y)(x - y)). -
Imaginary Numbers:
Defined by the square root of negative one, represented as (i), where (i = \sqrt{-1}). -
Sum of Squares:
Representation involving squares of numbers, though it cannot be factored into real numbers. Formula: (x^2 + y^2 = (x^2 + xy + y^2)). -
Distance Formula:
Used to calculate the distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)). Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}). -
Midpoint Formula:
Determines the midpoint between two points. Formula: (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)). -
Symmetry in Graphs:
- X-axis: A graph is symmetric with respect to the x-axis when ((x, y)) corresponds to ((x, -y)).
- Y-axis: Symmetry around the y-axis occurs when ((x, y)) corresponds to ((-x, y)).
- Origin: Symmetry about the origin is present when ((x, y)) corresponds to ((-x, -y)).
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Completing the Square:
A method to transform quadratic equations into vertex form. Formula: (x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2). -
Principal Square Root of a Number:
For a negative number (-a), the square root is defined in terms of imaginary numbers. Formula: (\sqrt{-a} = \sqrt{ai}). -
Slope of a Line:
The slope (m) of a line passing through two points can be calculated. Formula: (m = \frac{y_2 - y_1}{x_2 - x_1}). -
Parallel Lines:
Two lines are parallel if their slopes are equal, represented as (m_1 = m_2). -
Perpendicular Lines:
Two lines are perpendicular if their slopes are negative reciprocals, represented as (m_1 = -\frac{1}{m_2}). -
Even Function:
A function (f) is even if it satisfies (f(-x) = f(x)). -
Odd Function:
A function (f) is odd if it satisfies (f(-x) = -f(x)). -
Greatest Integer Function:
Often referred to as the "floor function," visualized like a slanted ladder. -
Vertical Shifts:
- Upward Shift: (h(x) = f(x) + c).
- Downward Shift: (h(x) = f(x) - c).
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Horizontal Shifts:
- Right Shift: (h(x) = f(x - c)).
- Left Shift: (h(x) = f(x + c)).
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Reflections:
- In the x-axis: (h(x) = -f(x)).
- In the y-axis: (h(x) = f(-x)).
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Operations on Functions:
- Sum: ((f + g)(x) = f(x) + g(x)).
- Difference: ((f - g)(x) = f(x) - g(x)).
- Product: ((fg)(x) = f(x)g(x)).
- Quotient: ((f / g)(x) = \frac{f(x)}{g(x)}).
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Composition of Functions:
Describes the output when a function (g) is applied to the input of another function (f). Formula: ((f \circ g)(x) = f(g(x))).
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Test your knowledge of key Algebra II formulas with this set of flashcards. Each card presents a crucial mathematical concept, such as point-slope form and the quadratic formula, making it a perfect tool for review and practice. Strengthen your understanding of algebraic principles effectively!
